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Chapter 3: ADC and DAC

Selecting The Antialias Filter

Table 3-2 summarizes the characteristics of these three filters, showing how
each optimizes a particular parameter at the expense of everything else. The
Chebyshev optimizes the roll-off, the Butterworth optimizes the passband
flatness, and the Bessel optimizes the step response.

The selection of the antialias filter depends almost entirely on one issue: how
information is represented in the signals you intend to process. While

there are many ways for information to be encoded in an analog waveform, only
two methods are common, time domain encoding, and frequency domain
encoding. The difference between these two is critical in DSP, and will be a
reoccurring theme throughout this book.

In frequency domain encoding, the information is contained in sinusoidal waves
that combine to form the signal. Audio signals are an excellent example of this.
When a person hears speech or music, the perceived sound depends on the
frequencies present, and not on the particular shape of the waveform. This can
be shown by passing an audio signal through a circuit that changes the phase of
the various sinusoids, but retains their frequency and amplitude. The resulting
signal looks completely different on an oscilloscope, but sounds identical. The
pertinent information has been left intact, even though the waveform has been
significantly altered. Since aliasing misplaces and overlaps frequency
components, it directly destroys information encoded in the frequency domain.
Consequently, digitization of these signals usually involves an antialias filter
with a sharp cutoff, such as a Chebyshev, Elliptic, or Butterworth. What about
the nasty step response of these filters? It doesn't matter; the encoded
information isn't affected by this type of distortion.

In contrast, time domain encoding uses the shape of the waveform to store
information. For example, physicians can monitor the electrical activity of a
person's heart by attaching electrodes to their chest and arms (an
electrocardiogram or EKG). The shape of the EKG waveform provides the
information being sought, such as when the various chambers contract during
a heartbeat. Images are another example of this type of signal. Rather than a
waveform that varies over time, images encode information in the shape of a
waveform that varies over distance. Pictures are formed from regions of
brightness and color, and how they relate to other regions of brightness and
color. You don't look at the Mona Lisa and say, "My, what an interesting
collection of sinusoids."

Here's the problem: The sampling theorem is an analysis of what happens in the
frequency domain during digitization. This makes it ideal to under-stand the
analog-to-digital conversion of signals having their information encoded in the
frequency domain. However, the sampling theorem is little help in
understanding how time domain encoded signals should be digitized. Let's take
a closer look.

Figure 3-15 illustrates the choices for digitizing a time domain encoded signal.
Figure (a) is an example analog signal to be digitized. In this case, the
information we want to capture is the shape of the rectangular pulses. A short
burst of a high frequency sine wave is also included in this example signal.
This represents wideband noise, interference, and similar junk that always
appears on analog signals. The other figures show how the digitized signal
would appear with different antialias filter options: a Chebyshev filter, a Bessel
filter, and no filter.

It is important to understand that none of these options will allow the original
signal to be reconstructed from the sampled data. This is because the original
signal inherently contains frequency components greater than one-half of the
sampling rate. Since these frequencies cannot exist in the digitized signal, the
reconstructed signal cannot contain them either. These high frequencies result
from two sources: (1) noise and interference, which you would like to eliminate,
and (2) sharp edges in the waveform, which probably contain information you
want to retain.

The Chebyshev filter, shown in (b), attacks the problem by aggressively
removing all high frequency components. This results in a filtered analog
signal that can be sampled and later perfectly reconstructed. However, the
reconstructed analog signal is identical to the filtered signal, not the original
signal. Although nothing is lost in sampling, the waveform has been severely
distorted by the antialias filter. As shown in (b), the cure is worse than the
disease! Don't do it!

The Bessel filter, (c), is designed for just this problem. Its output closely
resembles the original waveform, with only a gentle rounding of the edges. By
adjusting the filter's cutoff frequency, the smoothness of the edges can be traded
for elimination of high frequency components in the signal. Using more poles
in the filter allows a better tradeoff between these two parameters. A common
guideline is to set the cutoff frequency at about one-quarter of the sampling
frequency. This results in about two samples along the rising portion of each
edge. Notice that both the Bessel and the Chebyshev filter have removed the
burst of high frequency noise present in the original signal.

The last choice is to use no antialias filter at all, as is shown in (d). This has the
strong advantage that the value of each sample is identical to the value of the
original analog signal. In other words, it has perfect edge sharpness; a change
in the original signal is immediately mirrored in the digital data. The
disadvantage is that aliasing can distort the signal. This takes two different
forms. First, high frequency interference and noise, such as the example
sinusoidal burst, will turn into meaningless samples, as shown in (d). That is,
any high frequency noise present in the analog signal will appear as aliased
noise in the digital signal. In a more general sense, this is not a problem of the
sampling, but a problem of the upstream analog electronics. It is not the ADC's
purpose to reduce noise and interference; this is the responsibility of the analog
electronics before the digitization takes place. It may turn out that a Bessel
filter should be placed before the digitizer to control this problem. However,
this means the filter should be viewed as part of the analog processing, not
something that is being done for the sake of the digitizer.

The second manifestation of aliasing is more subtle. When an event occurs in
the analog signal (such as an edge), the digital signal in (d) detects the change
on the next sample. There is no information in the digital data to indicate what
happens between samples. Now, compare using no filter with using a Bessel
filter for this problem. For example, imagine drawing straight lines between
the samples in (c). The time when this constructed line crosses one-half the
amplitude of the step provides a subsample estimate of when the edge occurred
in the analog signal. When no filter is used, this subsample information is
completely lost. You don't need a fancy theorem to evaluate how this will affect
your particular situation, just a good understanding of what you plan to do with
the data once is it acquired.